User menu

User login

You are here

Primary tabs

type of a distribution function

Two distribution functionsF,G:ℝ→[0,1]normal-:FGnormal-→ℝ01F,G:\mathbb{R}\to[0,1] are said to of the same type if there exist a,b∈ℝabℝa,b\in\mathbb{R} such that G⁢(x)=F⁢(a⁢x+b)GxFaxbG(x)=F(ax+b). aaa is called the scale parameter, and bbb the location parameter or centering parameter. Let’s write F=tGsuperscripttFGF\stackrel{t}{=}G to denote that FFF and GGG are of the same type.

Remarks.

Necessarily a>0a0a>0, for otherwise at least one of G⁢(-∞)=0G0G(-\infty)=0 or G⁢(∞)=1G1G(\infty)=1 would be violated.

If G⁢(x)=F⁢(x+b)GxFxbG(x)=F(x+b), then the graph of GGG is shifted to the right from the graph of FFF by bbbunits, if b>0b0b>0 and to the left if b<0b0b<0.

If G⁢(x)=F⁢(a⁢x)GxFaxG(x)=F(ax), then the graph of GGG is stretched from the graph of FFF by aaa units if a>1a1a>1, and compressed if a<1a1a<1.

If XXX and YYY are random variables whose distribution functions are of the same type, say, FFF and GGG respectively, and related by G⁢(x)=F⁢(a⁢x+b)GxFaxbG(x)=F(ax+b), then XXX and a⁢Y+baYbaY+b are identically distributed, since

When XXX and a⁢Y+baYbaY+b are identically distributed, we write X=tYsuperscripttXYX\stackrel{t}{=}Y.

Again, suppose XXX and YYY correspond to FFF and GGG, two distribution functions of the same type related by G⁢(x)=F⁢(a⁢x+b)GxFaxbG(x)=F(ax+b). Then it is easy to see that E⁢[X]<∞EXE[X]<\inftyiffE⁢[Y]<∞EYE[Y]<\infty. In fact, if the expectation exists for one, then E⁢[X]=a⁢E⁢[Y]+bEXaEYbE[X]=aE[Y]+b. Furthermore, V⁢a⁢r⁢[X]VarXVar[X] is finite iff V⁢a⁢r⁢[Y]VarYVar[Y] is. And in this case, V⁢a⁢r⁢[X]=a2⁢V⁢a⁢r⁢[Y]VarXsuperscripta2VarYVar[X]=a^{2}Var[Y]. In general, convergence of moments is a “typical” property.

By the same token, we can classify all real random variables defined on a fixedprobability space according to their distribution functions, so that if XXX and YYY are of the same type ττ\tau iff their corresponding distribution functions FFF and GGG are of type ττ\tau.

Given an equivalence class of distribution functions belonging to a certain type ττ\tau, such that a random variable YYY of type ττ\tau exists with finite expectation and variance, then there is one distribution function FFF of type ττ\tau corresponding to a random variable XXX such that E⁢[X]=0EX0E[X]=0 and V⁢a⁢r⁢[X]=1VarX1Var[X]=1. FFF is called the standard distribution function for type ττ\tau. For example, the standard (cumulative) normal distribution is the standard distribution function for the type consisting of all normal distribution functions.

Within each type ττ\tau, we can further classify the distribution functions: if G⁢(x)=F⁢(x+b)GxFxbG(x)=F(x+b), then we say that GGG and FFF belong to the same location family under ττ\tau; and if G⁢(x)=F⁢(a⁢x)GxFaxG(x)=F(ax), then we say that GGG and FFF belong to the same scale family (under ττ\tau).